- The Discrete Schwarz-Pick Lemma Revisted
- Project Year:
2025
- REU Student (s):
Arham Lodha | The University of Texas at Austin TX
- Student 1 Institution:
The University of Texas at Austin
- Project Mentor:
Feng Luo
- Project Mentor Area:
Mathematics
- Project Abstract:
The Discrete Schwarz-Pick Lemma is a discrete analogue of the classical result from complex analysis, arising from the connection between circle packings and conformal maps established by Thurston. Previous works by Beardon-Stephanson and Van Eeuwen proved this lemma for circle packings where circles are tangent or intersect at non-obtuse angles, corresponding to inversive distances $I in [0,1]$. This paper extends the investigation to circle packings with obtuse intersections ($I in (-1,0)$) and disjoint packings ($I>1$). We prove that the Discrete Schwarz-Pick Lemma holds for the full range of intersecting circle packings with inversive distances in $(-1,1]$, provided an additional condition on the weights of each triangle is satisfied. The proof relies on a variational principle for circle packings with inversive distances. Conversely, we show that the lemma fails for disjoint circle packings where I≥1. This is demonstrated by constructing a specific counterexample on a triangulated disk with four vertices and providing a numerical analysis to confirm its validity beyond computational error.
